1. School of Physical Science and Information Engineerig, Liaocheng University, Liaocheng 252059, China 2. Shandong Provincial Key Laboratory of Optical Communication Science and Technology, Liaocheng University, Liaocheng 252059, China 3. Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, College of Physics and Engineering, Qufu Normal University, Qufu 273165, China
Extending the recent work completed by Fan et al. [Front. Phys. 9(1), 74 (2014)] to a two-mode case, we investigate how a two-mode squeezed vacuum evolves when it undergoes a two-mode amplitude dissipative channel, with the same decay rate κ, using the continuous-variable entangled state approach. Our analytical results show that the initial pure-squeezed vacuum state evolves into a definite mixed state with entanglement and squeezing, decaying over time as a result of amplitude decay. We also investigate the time evolutions of the photon number distribution, the Wigner function, and the optical tomogram in this channel. Our results indicate that the evolved photon number distribution is related to Jacobi polynomials, the Wigner function has a standard Gaussian distribution (corresponding to the vacuum) at long periods, losing its nonclassicality due to amplitude decay, and a larger squeezing leads to a longer decay time.
C. W. Gardner and P. Zoller, Quantum Noise, Berlin: Spinger, 2000
2
X. Y. Chen, Simultaneous amplitude and phase damping of x–psymmetric Gaussian states and their separability, Phys. Rev. A 73(2), 022307 (2006) https://doi.org/10.1103/PhysRevA.73.022307
3
D. Boyanovsky, Effective field theory during inflation: Reduced density matrix and its quantum master equation, Phys. Rev. D 92(2), 023527 (2015) https://doi.org/10.1103/PhysRevD.92.023527
4
L. Ferialdi, Exact non-Markovian master equation for the spin-boson and Jaynes-Cummings models, Phys. Rev. A 95(2), 020101(R) (2017)
5
H. Y. Fan and L. Y. Hu, Operator-sum representation of density operators as solutions to master equations obtained via the entangled state approach, Mod. Phys. Lett. B 22(25), 2435 (2008) https://doi.org/10.1142/S0217984908017072
6
Y. Takahashi and H. Umezawa, Thermo field dynamics, Collect. Phenom. 2, 55 (1975)
7
H. Y. Fan and L. Y. Hu, New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation, Opt. Commun. 281(22), 5571 (2008) https://doi.org/10.1016/j.optcom.2008.08.002
8
X. G. Meng, H. S. Goan, J. S. Wang, and R. Zhang, Nonclassical thermal-state superpositions: Analytical evolution law and decoherence behavior, Opt. Commun. 411(3), 15 (2018) https://doi.org/10.1016/j.optcom.2017.11.005
9
X. G. Meng, Z. Wang, H. Y. Fan, and J. S. Wang, Nonclassicality and decoherence of photon-subtracted squeezed vacuum states, J. Opt. Soc. Am. B 29(11), 3141 (2012) https://doi.org/10.1364/JOSAB.29.003141
10
H. Y. Fan, S. Wang, and L. Y. Hu, Evolution of the singlemode squeezed vacuum state in amplitude dissipative channel, Front. Phys. 9(1), 74 (2014) https://doi.org/10.1007/s11467-013-0367-z
C. J. Liu, W. Ye, W. D. Zhou, H. L. Zhang, J. H. Huang, and L. Y. Hu, Entanglement of coherent superposition of photon-subtraction squeezed vacuum, Front. Phys. 12(5), 120307 (2017) https://doi.org/10.1007/s11467-017-0694-6
P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit, Phys. Rev. Lett. 104(10), 103602 (2010) https://doi.org/10.1103/PhysRevLett.104.103602
15
H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, Quantum repeaters: The role of imperfect local operations in quantum communication, Phys. Rev. Lett. 81(26), 5932 (1998) https://doi.org/10.1103/PhysRevLett.81.5932
16
F. Jia, S. Xu, C. Z. Deng, C. J. Liu, and L.Y. Hu, L. Y. Hu, 3D entangled fractional squeezing transformation and its quantum mechanical correspondence, Front. Phys. 11(3), 110302 (2016) https://doi.org/10.1007/s11467-015-0538-1
17
H. Y. Fan and Y. Fan, New representation for thermo excitation and de-excitation in thermofield dynamics,Phys. Lett. A 282(4–5), 269 (2001) https://doi.org/10.1016/S0375-9601(01)00198-0
18
K. M. Zheng, S. Y. Liu, H. L. Zhang, C. J. Liu, and L. Y. Hu, A generalized two-mode entangled state: Its generation, properties, and applications, Front. Phys. 9(4), 451 (2014) https://doi.org/10.1007/s11467-014-0419-z
19
N. R. Zhou, J. F. Li, Z. B. Yu, L. H. Gong, and A. Farouk, New quantum dialogue protocol based on continuous-variable two-mode squeezed vacuum states, Quantum Inform. Process. 16(1), 4 (2017) https://doi.org/10.1007/s11128-016-1461-2
20
Z. X. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, Adaptive phase estimation with two-mode squeezed vacuum and parity measurement, Phys. Rev. A 95(5), 053837 (2017) https://doi.org/10.1103/PhysRevA.95.053837
21
T. Opatrný, G. Kurizki, and D. G. Welsch, Improvement on teleportation of continuous variables by photon subtraction via conditional measurement, Phys. Rev. A 61(3), 032302 (2000) https://doi.org/10.1103/PhysRevA.61.032302
22
J. P. Olson, K. P. Seshadreesan, K. R. Motes, P. P. Rohde, and J. P. Dowling, Sampling arbitrary photonadded or photon-subtracted squeezed states is in the same complexity class as boson sampling, Phys. Rev. A 91(2), 022317 (2015) https://doi.org/10.1103/PhysRevA.91.022317
23
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Berlin: Springer, 1966 https://doi.org/10.1007/978-3-662-11761-3
24
H. Y. Fan and J. R. Klauder, Eigenvectors of two particles’ relative position and total momentum, Phys. Rev. A 49(2), 704 (1994) https://doi.org/10.1103/PhysRevA.49.704
25
Y. Z. Li, F. Jia, H. L. Zhang, J. H. Huang, and L. Y. Hu, Hermite polynomial excited squeezed vacuum as quantum optical vortex states, Laser Phys. Lett. 12(11), 115203 (2015) https://doi.org/10.1088/1612-2011/12/11/115203
26
S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs, Laser Phys. Lett. 12(4), 045201 (2015) https://doi.org/10.1088/1612-2011/12/4/045201
27
H. L. Zhang, H. C. Yuan, L. Y. Hu, and X. X. Xu, Synthesis of Hermite polynomial excited squeezed vacuum states from two separate single-mode squeezed vacuum states, Opt. Commun. 356(12), 223 (2015) https://doi.org/10.1016/j.optcom.2015.07.083
A. Wünsche, About integration within ordered products in quantum optics, J. Opt. B Quantum Semiclassical Opt. 1(3), R11 (1999) https://doi.org/10.1088/1464-4266/1/3/201
30
H. Y. Fan and H. R. Zaidi, Application of IWOP technique to the generalized Weyl correspondence, Phys. Lett. A 124(6–7), 303 (1987) https://doi.org/10.1016/0375-9601(87)90016-8
31
J. S. Wang, H. Y. Fan, and X. G. Meng, A generalized Weyl–Wigner quantization scheme unifying P-Qand Q-Pordering and Weyl ordering of operators, Chin. Phys. B 21(6), 064204 (2012) https://doi.org/10.1088/1674-1056/21/6/064204
32
W. Schleich and J. A. Wheeler, Oscillations in photon distribution of squeezed states and interference in phase space, Nature 326(6113), 574 (1987) https://doi.org/10.1038/326574a0
33
L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, Photonsubtracted squeezed thermal state: Nonclassicality and decoherence, Phys. Rev. A 82(4), 043842 (2010) https://doi.org/10.1103/PhysRevA.82.043842
34
H. Y. Fan, Entangled states, squeezed states gained via the route of developing Dirac’s symbolic method and their applications, Int. J. Mod. Phys. B 18(10&11), 1387 (2004) https://doi.org/10.1142/S0217979204024835
35
X. G. Meng, J. S. Wang, and B. L. Liang, A new finitedimensional pair coherent state studied by virtue of the entangled state representation and its statistical behavior, Opt. Commun. 283(20), 4025 (2010) https://doi.org/10.1016/j.optcom.2010.05.058
36
X. G. Meng, Z. Wang, J. S. Wang, and H. Y. Fan, Wigner function, optical tomography of two-variable Hermite polynomial state, and its decoherence effects studied by the entangled-state representations, J. Opt. Soc. Am. B 30(6), 1614 (2013) https://doi.org/10.1364/JOSAB.30.001614
37
H. Y. Fan, J. H. Chen, and P. F. Zhang, On the entangled fractional squeezing transformation, Front. Phys. 10(2), 187 (2015) https://doi.org/10.1007/s11467-014-0457-6
38
H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J. Opt. B Quantum Semiclassical Opt. 5(4), R147 (2003) https://doi.org/10.1088/1464-4266/5/4/201
39
H. Y. Fan and G. C. Yu, Radon transformation of the Wigner operator for two-mode correlated system in generalized entangled state representation, Mod. Phys. Lett. B 15(07), 499 (2000) https://doi.org/10.1142/S0217732300000487
